A273556 Decimal expansion of Rosser's constant.

8, 3, 2, 4, 2, 9, 0, 6, 5, 6, 6, 1, 9, 4, 5, 2, 7, 8, 0, 3, 0, 8, 0, 5, 9, 4, 3, 5, 3, 1, 4, 6, 5, 5, 7, 5, 0, 4, 5, 4, 4, 5, 3, 1, 8, 0, 7, 7, 4, 1, 7, 0, 5, 3, 2, 4, 0, 8, 9, 3, 9, 9, 1, 2, 9, 6, 0, 3, 4, 7, 0, 7, 1, 3, 9, 4, 8, 1, 1, 4, 2, 4, 2, 1, 9, 1, 6, 2, 7, 2, 2, 5, 0, 4, 6, 3, 8, 1

Offset: 0

Author

Jean-François Alcover, May 25 2016

Keywords

• nonn
• cons

Comments

Named after the American logician and mathematician John Barkley Rosser, Sr. (1907-1989). - Amiram Eldar, Jun 20 2021

Examples

			0.832429065661945278030805943531465575045445318077417053240893991296...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood constants, p. 86.

Links

• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., Vol. 6, No. 1 (1962), pp. 64-94, eq. (2.14).
• Eric Weisstein's World of Mathematics, Twin Primes Constant.

Crossrefs

Cf. A005597, A091724, A246061.

Programs

Mathematica

digits = 98; s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 60]; C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n) )*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 60}];
RealDigits[4*C2/Exp[2*EulerGamma], 10, digits] // First

PARI

4 * exp(-2*Euler) * prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 17 2021

Formula

4*C_2/exp(2*EulerGamma), where C_2 is the twin primes constant.

Equals lim_{x->inf} Product_{2 < p <= x} (1-2/p)*log(x)^2.